A geometric proof of the upper bound on the size of partial spreads in H(4n+1, q²)

We give a geometric proof of the upper bound of q(2n+1) + 1 on the size of partial spreads in the polar space H(4n + 1, q(2)). This bound is tight and has already been proved in an algebraic way. Our alternative proof also yields a characterization of the partial spreads of maximum size in H(4n + 1, q(2))

The maximum size of a partial spread in H(4n +1,q²) is q^(2 n +1) + 1

We prove that in every finite Hermitian polar space of odd dimension and even maximal dimension rho of the totally isotropic subspaces, a partial spread has size at most q(rho+1) + 1, where GF(q(2)) is the defining field. This bound is tight and is a generalisation of the result of De Beule and Metsch for the case rho = 2

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch

Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes

On Q-polynomial regular near 2d-gons

We discuss thick regular near 2d-gons with a Q-polynomial collinearity graph. For da parts per thousand yen4, we show that apart from Hamming near polygons and dual polar spaces there are no thick Q-polynomial regular near polygons. We also show that no regular near hexagons exist with parameters (s, t (2), t) equal to (3, 1, 34), (8, 4, 740), (92, 64, 1314560), (95, 19, 1027064) or (105, 147, 2763012). Such regular near hexagons are necessarily Q-polynomial. All these nonexistence results imply the nonexistence of distance-regular graphs with certain classical parameters. We also discuss some implications for the classification of dense near polygons with four points per line

Theorems of Erdős–Ko–Rado type in polar spaces

AbstractWe consider Erdős–Ko–Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect non-trivially. We characterize the Erdős–Ko–Rado sets of generators of maximum size in all polar spaces, except for H(4n+1,q2) with n⩾2

Carnosinase-1 overexpression, but not aerobic exercise training, affects the development of diabetic nephropathy in BTBR ob/ob mice

Manipulation of circulating histidine-containing dipeptides (HCD) has been shown to affect the development of diabetes and early-stage diabetic nephropathy (DN). The aim of the present study was to investigate whether such interventions, which potentially alter levels of circulating HCD, also affect the development of advanced-stage DN. Two interventions, aerobic exercise training and overexpression of the human carnosinase-1 (hCN1) enzyme, were tested. BTBR ob/ob mice were either subjected to aerobic exercise training (20 wk) or genetically manipulated to overexpress hCN1, and different diabetes- and DN-related markers were compared with control ob/ob and healthy (wild-type) mice. An acute exercise study was performed to elucidate the effect of obesity, acute running, and hCN1 overexpression on plasma HCD levels. Chronic aerobic exercise training did not affect the development of diabetes or DN, but hCN1 overexpression accelerated hyperlipidemia and aggravated the development of albuminuria, mesangial matrix expansion, and glomerular hypertrophy of ob/ob mice. In line, plasma, kidney, and muscle HCD were markedly lower in ob/ob versus wild-type mice, and plasma and kidney HCD in particular were lower in ob/ob hCN1 versus ob/ob mice but were unaffected by aerobic exercise. In conclusion, advanced glomerular damage is accelerated in mice overexpressing the hCN1 enzyme but not protected by chronic exercise training. Interestingly, we showed, for the first time, that the development of DN is closely linked to renal HCD availability. Further research will have to elucidate whether the stimulation of renal HCD levels can be a therapeutic strategy to reduce the risk for developing DN